Probabilistic approach for comparing first eigenvalues

A NNALES DE L ’I. H. P., SECTION B

C ^RISTINA B ^ETZ H ^ENRYK G ^ZYL

Probabilistic approach for comparing first eigenvalues

Annales de l’I. H. P., section B, tome 21, n

^o

2 (1985), p. 147-156

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Probabilistic approach

for comparing ^first eigenvalues

Cristina BETZ

(*) and ^Henryk

^GZYL

^(*)

A. P. 52120, Caracas 1050-A, ^Venezuela Ann. Inst. Henri Poincaré,

Vol. 21, ^n" 2, 1985, p. 147-156. Probabilités et Statistiques

ABSTRACT. ^- We ^use

probabilistic

subordination and time

change

^to

compare the first

eigenvalues

^for

problems

^{of the}

type G~

⁼

0 ;

+ ⁼

0,

^{with zero}

boundary

conditions.

and ph~-ases : ^First eigenvalue, subordinate _process, time change.

RESUME. ^- Nous utilisons la subordination

stochastique

^{et les chan-}

gements

^de

temps

pour comparer les

premières

valeurs propres des pro- blemes du

type

^{+ =}

0 ; qyl

^{+ =}

0,

^avec des conditions frontières nulles.

1. INTRODUCTION

The aim of this paper is ^{to use}

probabilistic

subordination and time

change

^to compare the first

eigenvalues

^{of the}

problems

for an

adequate

^{class of}

operators

^G.

(*) Permanent adress: Apartado 52120. Caracas 1050-A Venezuela. Work done during

a sabbatical stay ^{at the} Mathematical Centre.

Annales de I’Institut Henri Poincaré - Probabilités et Statistiques - ^Vol. 21 _ 0246-0203 85/02/147/ 10/S 3.00j~.c~ Gauthier-Villars

147

148 ^C. BETZ AND H. GZYL

In Section 2 we consider a Hunt _process

(Xt )

^{with state} space

(E, ~)

whose

semigroup

possesses ^a

nonnegative symmetric density f (t, x, y)

with

respect

^{to a Radon} measure m on

(E, ~),

^{such that if D is a bounded}

open subset of E with ⁼ 0 then

and such

that,

^{if T =}

inf { t

^{> 0 :}

^Xt

^E

D~ ~

is the first exit time from

D,

^then

By stopping (Xt)

^{at the}

boundary

^{of D and}

subjecting

^{it to a} local death

rate

q(x)

^{we are able to} compare the first

eigenvalues

^{of the}

problems (1.1)

and

(1.2),

^{when G is the} infinitesimal

generator

^of

(Xt).

In Section 3 we consider an

adequate

^time

change

^of the process with

generator G - q

^to compare the first

eigenvalues

^of

problems (1. 2)

^and

(1. 3).

Basic for all the

comparison

^{results is the fact that under the stated}

assumptions

^{on the}

semigroup

^{one can}

get eigenvalue expansions

^{for these}

various _processes which allow one to

give

^a

simple

characterization of the first

eigenvalue.

This characterization

(proposition

^{1 in Section}

2)

^is

well known for diffusions

(see chapter

^{14 of}

[4 ]).

Section 4 is devoted to the discussion of

examples.

We refer the reader to

[7] ]

^{for the basic notation} and definitions

throughout

the paper.

2. COMPARISON OF

( 1.1 )

^AND

(1.2)

Let

(XP)

^be the process

(Xt)

^{killed when} it leaves D and

(QP)

^{its tran-}

sition

semigroup,

^{that is}

It is

proved

ⁱⁿ

[5-] that

^{there is a}

sequence { ~,~ ~

^of

eigenvalues

^{of G such}

that 0

~.1 ~,~

^... ..., -~ oo and a

complete

^ortho-

normal set of

eigenfunctions ~ c~ J ~

such that the series converges

absolutely

^and

for

L :leD).

ⁿ

Annales de I’Institut Henri Poincaré - Probabilités et Statistiques

COMPARING FIRST EIGENVALUES 149

A similar

expansion

^{can be}

^found

^{for the}

semigroup

of the process subordinate to

(X?)

^with

^respect

^{to the}

multiplicative

^func-

tional

Mt

⁼⁼

2014 ~o ^{where ~}

^{is a} continuous

positive

^function

on E

(see

^also

[J ]).

The first

eigenvalue

^{of the}

problem

can now be characterized as follows.

PROPOSITION

^{1. -}

If ~.1

^{is the}

first eigenvalue of

^the

prob lem (1.1 )

^then

Proof

^{It follows from}

(2.1)

^that

Let

K,~ _ l~

^D

^y)m(dy)

^{then one has}

Clearly,

^{when ~, ~}

~,1

this series

diverges. Also,

^E"

] is

^an

increasing

function of

h,

^so it suffices to show that E"

]

^{oo for ~, Vx E D.}

Now,

^{if 0 A}

~,1

^then

hence

REMARK 1. ^- A similar characterization for the first

eigenvalue

^of

problem (1.2) clearly

^holds.

Vol. 21, ^n° 2-1985.

150 C. BETZ AND H. GZYL

REMARK 2. ^- The heuristics of the situation is as follows :

So what one does is ^« look at the

poles

of the resolvent ».

In order to compare the first

eigenvalues

^of

problems (1.1)

^and

(1.2)

we consider

(Xt ),

the canonical realization of the process subordinated to

(Xt)

^with

respect to Mt

⁼ exp - ^on the

space Q

^{= Q x}

[0,

^oo

],

and denote

by

^{Px its}

corresponding

^{measures on Q}

(see [1 ],

^{ch. 3 for all}

the

constructions).

Following [1 ],

^let y : ~

[o, oo ]

^{be the}

projection y(co, 03BB)

⁼

03BB,

^and

LEMMA 1. ^-

] _

^Ex

[e~T ],

^{‘dx E D.}

Proof

^{- It} follows from the aefinition of

(Xt)

^that

N

and

therefore, by

^the definition of x we have

Now

the

integration by parts

in the third

equality

^is

possible

^since

P"(T ~)

⁼¹

’dxED.

The

following comparison

^{result now follows}

easily

^from

Proposition

¹

and Lemma 1.

N

THEOREM 1. ^- ~et

~l

^{be the smallest}

eigenvalue of problem (1.1)

^and

~.

the smallest

eigenvalae of problem (1.2}

^then

Annales de l’lnstitut Henri Poincaré - Probabilités et Statistiques

COMPARING FIRST EIGENVALUES 151

REMARK 3. ^- In the same manner, ^{one can} compare the first

eigenvalues

of the

problems

for

q2 >_

q1. In

fact,

^let

(Xt)

^now denote the process with generator and let

(Xt)

be the process subordinated to

(Xt)

^with

respect

^to

then,

^{the same}

reasoning

^{as before}

gives

REMARK 4. ^- The

comparison

results will hold whenever a characte- rization for the

principal eigenvalue,

^{as the one}

given

ⁱⁿ

proposition

¹

holds; therefore,

^{these results} will be valid for the

elliptic operators

^consi- dered

by

^Friedman

[4 ~,

^Vol.

2, Chapter

^14.

3. COMPARISON OF

(1.2)

^AND

(1.3)

We will now compare the first

eigenvalues

^{of the}

problems

for p continuous and _{p >} c > 0.

Let

(Xt)

^{now be} the process with

generator G - q

^{= G}

(we

^{warn the}

reader that to be consistent with the notation of section 2 we should work with

(Xt),

^{but we feel} this would become too

confusing).

Suppose (Xt)

^{has a}

symmetric

transition

density pet, x, y)

^with

respect

to the Radon measure m which satisfies

for bounded ^sets D.

Let

(At)

^{be the}

following

^additive functional of

(XJ

Vol. 21, ^n° 2-1985.

152 C. BETZ AND H. GZYL

and let

(Lt)

^{be the}

right

continuous inverse of

(At);

^{that is}

One may check that satisfies

Call

~~

⁼

_X1:t’

^then

X - (Q, ~ , ~~t, Xt, 8~t, Px)

^{is a}

strong

^Markov process

(see [1 ],

^Ch.

V)

^{and it}

follows

^from

Dynkin’s

^{formula that}

(Xt)

has gene- rator - .

P

The

following proposition

^{allows one to use the same}

procedure

^as

that of

proposition

¹ in Section 2 to characterize the first

eigenvalue

^of

problem (1. 3)

^{as N}

~- , ~ - - T - __ ,

where

PROPOSITION 1. ^- Under the

assumptions (3 .1 )

^and

(3 . 2)

^{and the} sym- metry

of p(t,

x,

y);

^the process

(Xt)

possesses ^a

symmetric

transition

density p{t,

^x,

y)

^with respect ^to the measure

p( y)m{dy)

^and moreover,

if D

^{is bounded}

then

Proof.

^{- Observe that}

where

If

(Xt)

^{has a}

symmetric

transition

density p(t, x, y)

^then

(Qf)

^{has a}

^symmetric

transition

density

^{as well}

(see [5 ]),

^{call it}

q°‘(t,

x,

y).

^{It is} easy ^{to see} that

(recall

^c is such that

p(x)

^{> c >}

0).

Annales de l’Institut Henri Poincaré - Probabilités et Statistiques

153 COMPARING FIRST EIGENVALUES

Let

u° (x, y)

^{= x,}

y)ds,

^then

by inverting Laplace

transforms we 0

get,

^{for an}

appropriate

^contour y

which proves ^the existence of the

symmetric

transition

density fi(t, x, y).

To prove

(3.3)

^it is enough to show that

But,

for all a > 0

Hence

and the result follows from

(3.2).

Let

then

and so

(see [1 ],

^ch.

V).

THEOREM l. ^-

If

p 1 then the

first eigenvalue of problem (1.3)

^is

bigger

^{than the}

first eigenvalue of problem (1.2)

^and

if

p ^> 1 the

first eigen-

value

of problem (1.3)

^is smaller than the

first eigenvalue of problem (1.2).

Vol. 21, ^{n° 2-1985.}

154 C. BETZ AND H. GZYL

Proof

Notice that for

p > l,

^from

(3. 5)

^{it follows that}

4. EXAMPLES

In this section we list some situations in which the

comparison

^results

hold. In

particular,

^{we see that the classical} results of

examples

^{1 and 2}

(see [3 ], chap VI)

^{are valid for more}

general

situations.

EXAMPLE 1. The

eigenvalues

^{of the}

problem

ⁱⁿ

D,

~

^{= 0 on}

aD;

^increase

when q

^increases.

EXAMPLE 2. - The

eigenvalues

^{of +}

~,~

^{= 0 in}

^D, c~

^{= 0}

on

aD ;

^{increase when the domain D decreases.}

EXAMPLE 3. ^- This is a

generalization

^of

example

1. Let G be the

Laplace-Beltrami operator

^{on a} Riemannian manifold or

compact

^Lie group and D an open bounded subset. The same conclusions hold. See

[7]

or

[8 ]

^{for the}

appropriate

constructions.

EXAMPLE 4. - The

following construction, going

^{back to}

Phillips [6]

yields

^a

larger

^{class of} processes

(and eigenvalue problems)

^{for which the}

comparison

^results

apply. Suppose (Xt)

^{has a} transition

density p(t, x, y)

such that

where the are an

orthonormal, complete

set, ^and

~,,~ )

^{are all the}

solutions to +

~n -

⁰

plus appropriate boundary

conditions.

Let now denote a subordinator

(a

process with

stationary

^inde-

pendent

increments ^on

[0, oo ]) independent

^of

(Xt).

^{Define a new semi-}

group ^on the same state space

by

where Jln = a~,n - l)v(du), (a, v) being

the characteristics of

(~ t )..

Annales de l’Institut Henri Poincaré - Probabilités et Statistiques

155 COMPARING FIRST EIGENVALUES

In this _case, the _process

(XTt’ ^Q,

^where

^Pz

is constructed from

(Qt)

via the

Kolmogorov

^extension

theorem,

^has transition

semigroup (Qt)

and infinitesimal

generator

One has + ⁼ 0

and,

^from

completeness,

^there are no more

solutions to

G~ ~r + ~.~

^{= 0}

(with

^{the same}

boundary

conditions as for

=

0).

EXAMPLE 5. ^- The

comparison

^{results can be}

applied

^to the sym- metric stable process in of

index a,

^{0 a}

2 ;

^that

is,

the process with

stationary independent

increments whose continuous transition

density

relative to

Lebesgue

^{measure in}

^[RN

^is

See

[2 ] and [5 ] in

connection with this

example.

It is

interesting

^{to observe} the connection with

example 4,

^{which is as} follows : If

(Nf)

^{is the one sided stable}

^semigroup

ⁱⁿ

^(0,

^E

^{(0, 1)}

^and

A)

^the transition

semigroup

^{of the} Brownian motion process in

IRN then,

^the

semigroup

^{of the}

symmetric

^stable process in

(~~

satisfies

(see [1 ], ch. I ; 2 . 20).

The _process Xa with

semigroup Pt given

^{above can} be described ^as

X? = B(Nf), a/2 = ~3, B(t ) being

^{Brownian motion on (~n and}

Nf

^{as above}

too. As in section 2

defined for

f

^{bounded or}

positive,

describes the

semigroup

^of the process obtained from ~~

by killing

upon

leaving

^a bounded open ^set D. T

being

the first

hitting

^{time of D~.}

Again,

the results of Getoor in

[5]

^are valid int this case as well and

we can therefore

apply

^the

comparison

^{theorem to a class of}

integral

^ope-

rators

(or integral equations)

^on bounded domains.

Actual

explicit computation

of the killed

semigroup

^{and its associated}

infinitesimal

generator

^{in the one} dimensional case can be seen in

[9 ] and

in the references listed there. The domain D

being

^{an interval on the line-or}

( - 1, 1)

^after

centering

^and

scaling.

Vol. 21, n° 2-1985.

156 C. BETZ AND H. GZYL

EXAMPLE 6. ^- The Ornstein-Uhlenbeck process in R has ^a transition

density f(t, x, y)

^with

respect

^{to the measure}

m(d y)

⁼

e -’’2/2 dy

^{which is}

symmetric

^see

[2 ].

EXAMPLE 7. - The

following

situation may

apply

^to discretized ^ver- sions of

examples

^{1 or} 2. Consider a

symmetric

Markov chain on a

lattice,

with transition _among nearest

neighbors only.

^{Let D a be subset of the}

state space and let aD denote the subset of D~

consisting

^{of nearest}

neighbors

to

points

^{in D.}

IfQ

denotes the rate matrix of the process killed ^on

leaving D,

then our

comparison

^result

applies

^{to the}

problems

ACKNOWLEDGEMENTS

We ^{want to} thank R. K. Getoor for his comments on the

preprint.

In the

original preprint, example

⁵ is wrong. He

pointed

^{it out for us.}

REFERENCES

[1] ^{R. M.} BLUMENTHAL and R. K. GETOOR, Markov processes and Potential theory, ^Aca-

demic Press, ^New York, ^1968.

[2] ^{R. M.} BLUMENTHAL and R. K. GETOOR, The asymptotic distribution of the eigenvalues

for a class of Markov operators, Pacific ^Jour. Math., ^t. 9, 1959, p. 399-408.

[3] R. COURANT and D. HILBERT, Methods of Mathematical physics, ^{t. I.} Interscience.

New York, 1953.

[4] ^A. FRIEDMAN, Stochastic differential equations ^and applications. ^Academic Press, New York, 1976.

[5] ^{R. K.} GETOOR, Markov operators ^and their associated semi-groups, Pacific ^Jour.

Jour. Math., ^t. 9, 1959, p. 449-472.

[6] ^{R. S.} PHILLIPS, ^{On the} generation ^of semigroups ^{of linear} operators, Pacific ^Jour.

Math., ^t. 2, 1952, p. 343-369.

[7] ^{M. A.} PINSKY, Isotropic transport process ^{on a} Riemannian manifold, ^{T. A. M.} S.,

t. 218, 1976, p. 353-360.

[8] ^{E. M.} STEIN, Topics ^{in harmonic} analysis, Annales of Math. Studies # 63. Princeton Univ. Press. Princeton N. J., ^1970.

[9] ^S. WATANABE, On stable _processes with boundary conditions, ^{Jour. Math. Soc.} Jap.,

t. M, n° 2, 1962, p. 170-198.

(Manuscrit reçu le 3 juillet 1984)

Annales de l’Institut Henri Poincaré - Probabilités et Statistiques